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Stability, Steady‐State Bifurcations, and Turing Patterns in a Predator–Prey Model with Herd Behavior and Prey‐taxis
Author(s) -
Song Yongli,
Tang Xiaosong
Publication year - 2017
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12165
Subject(s) - bifurcation , mathematics , stability (learning theory) , steady state (chemistry) , mathematical analysis , predation , neumann boundary condition , taxis , bifurcation theory , statistical physics , equilibrium point , boundary (topology) , control theory (sociology) , differential equation , physics , computer science , nonlinear system , ecology , chemistry , quantum mechanics , machine learning , transport engineering , engineering , biology , control (management) , artificial intelligence
In this paper, we consider a predator–prey model with herd behavior and prey‐taxis subject to the homogeneous Neumann boundary condition. First, by analyzing the characteristic equation, the local stability of the positive equilibrium is discussed. Then, choosing prey‐tactic sensitivity coefficient as the bifurcation parameter, we obtain a branch of nonconstant solutions bifurcating from the positive equilibrium by an abstract bifurcation theory, and find the stable bifurcating solutions near the bifurcation point under suitable conditions. We have shown that prey‐taxis can destabilize the uniform equilibrium and yields the occurrence of spatial patterns. Furthermore, some numerical simulations to illustrate the theoretical analysis are also carried out, Turing patterns such as spots pattern, spots–strip pattern, strip pattern, stable nonconstant steady‐state solutions, and spatially inhomogeneous periodic solutions are obtained, which also expand our theoretical results.